p-Adic L-Functions and Selmer Groups

This award will support the principal investigator’s research in number theory. A central focus of number theory is understanding the structure of rational solutions to polynomial equations with rational coefficients. Since the 20th century, L-functions—a special class of complex analytic functions defined through an infinite product over prime numbers—have emerged as an essential tool in advancing this understanding. There is a prevailing belief that deep connections exist between the arithmetic properties of polynomials equations and the behavior of their associated L-functions.The Bloch-Kato conjecture predicts such connections in a broad and unifying framework, and the Iwasawa-Greenberg main conjecture offers an analogue in the setting of p-adic deformations. The PI will investigate these conjectures. This award will also support the mentoring of undergraduate and graduate students, the organization of seminars and several outreach activities.

The PI’s research will largely focus on developing new techniques for studying p-adic properties of algebraic automorphic forms, which provide a useful bridge between the Selmer groups and L-functions for Galois representations arising from automorphic representations. The key technical components include: studying p-adic deformations of iterations of geometric Maass-Shimura differential operators for symplectic and unitary groups within the framework of classical and higher Coleman theory; generalizing the construction of p-adic L-functions for symplectic and unitary groups to cases where the ordinary locus of the associated Shimura varieties is empty, by analyzing the dynamics of Up operators; investigating the p-adic properties of various Eisenstein series on unitary and orthogonal groups to produce the inputs for bounding Selmer groups through congruences among automorphic forms and Euler systems; and constructing new Euler systems by producing extensions of Galois representations via étale cohomologies of Shimura varieties, and by using certain degenerate cases of Gross-Kudla-Schoen diagonal cycles. Built on the results obtained in these technical components, new cases of the Bloch-Kato conjecture and the Iwasawa-Greenberg main conjecture will be deduced.

This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

Award Number: 2501507
Principal Investigator: Zheng Liu
Funds Obligated: $125,986
State: CA